Paper 2018/648 Original: eprint.iacr.org/2018/648 ↗

Semi-Commutative Masking: A Framework for Isogeny-based Protocols, with an Application to Fully Secure Two-Round Isogeny-based OT

Abstract

We define semi-commutative invertible masking structures which aim to capture the methodology of exponentiation-only protocol design (such as discrete logarithm and isogeny-based cryptography). We discuss two instantiations: the first is based on commutative group actions and captures both the action of exponentiation in the discrete logarithm setting and the action of the class group of commutative endomorphism rings of elliptic curves, in the style of the CSIDH key-exchange protocol; the second is based on the semi-commutative action of isogenies of supersingular elliptic curves, in the style of the SIDH key-exchange protocol. We then construct two oblivious transfer protocols using this new structure and prove that these UC-securely realise the oblivious transfer functionality in the random-oracle-hybrid model against passive adversaries with static corruptions. Moreover, by starting from one of these two protocols and using the compiler introduced by Döttling et al. (Eurocrypt 2020), we achieve the first fully UC-secure two-round OT protocol based on supersingular isogenies.